Do they actually buy wholesale and stock inventory or is it more like an API dropship model? Especially for furniture since storing such inventory can be costly.
Generally it works the same as a physical showroom. Some pieces are stocked, and some are made-to-order. But if you don't have inventory, then you're subject to price shopping by customers who will go to the next place that does have inventory, or a better price with less wait. I can't think of any furniture store that is drop-shipping. There's very little incentive for furniture makers to be in the shipping business, one at a time. There aren't many furniture stores that are e-commerce only, most are a hybrid of physical showroom and e-commerce capabilities. But generally people buy furniture in person way more often, since it's one of those higher dollar items where the majority of people want to try for fit and comfort before buying, touching fabrics, confirming colors, etc. While there are still people who go into a showroom to browse and then try to buy online, the showrooms work hard to mask the model numbers making it far more difficult to price shop.
There's plenty of discussion about furniture sales. And as you've noted, the cost of showrooms and holding stock often drives new furniture companies out of business. If you plan to run a furniture business, you had better get your marketing down solid before you start building your store.
My recommendation is that you look at the largest furniture stores as a secure model more than the smallest.